Exact phase shifts for atom interferometry - CiteSeerX

For this purpose we derive two new theorems (the four end- points theorem and the phase shift formula) valid for a Hamiltonian at most quadratic in position and ...
109KB taille 2 téléchargements 306 vues
Physics Letters A 306 (2003) 277–284 www.elsevier.com/locate/pla

Exact phase shifts for atom interferometry Ch. Antoine a,∗ , Ch.J. Bordé a,b,∗ a Equipe de Relativité Gravitation et Astrophysique, LERMA, CNRS-Observatoire de Paris, Université Pierre et Marie Curie,

4 place Jussieu, 75005 Paris, France b Laboratoire de Physique des Lasers, UMR 7538 CNRS, Université Paris Nord, 99 avenue J.-B. Clément, 93430 Villetaneuse, France

Received 28 October 2002; accepted 8 November 2002 Communicated by P.R. Holland

Abstract In the case of an external Hamiltonian at most quadratic in position and momentum operators, we use the ABCDξ formulation of atom optics to establish an exact analytical phase shift expression for atom interferometers with arbitrary spatial or temporal beam splitter configurations. This result is expressed in terms of coordinates and momenta of the wave packet centers at the interaction vertices only.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction Recently atom interferometers [1] have been described by the ABCDξ formalism of Gaussian atom optics [2,3] which yields an exact formulation of phase shifts taking into account the wave packet structure of atom waves. For the theory of atom interferometers two basic stages are required: 1. A proper description of the propagation of wave packets between the beam splitters; 2. An adequate modelization of the beam splitters themselves. The first stage is achieved through the ABCDξ theorem whose main results are briefly recalled in Section 2. The second problem is addressed by the ttt theorem which provides a simple model for the phase introduced by the splitting process. In this Letter we give a compact way to express the atom interferometer phase shifts in terms of the coordinates and momenta of the wave packet centers only. For this purpose we derive two new theorems (the four endpoints theorem and the phase shift formula) valid for a Hamiltonian at most quadratic in position and momentum operators. * Corresponding authors.

E-mail addresses: [email protected] (Ch. Antoine), [email protected] (Ch.J. Bordé). 0375-9601/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(02)01625-0

278

Ch. Antoine, Ch.J. Bordé / Physics Letters A 306 (2003) 277–284

2. The ABCDξ theorem In this framework we consider a Hamiltonian which is the sum of an internal Hamiltonian H0 (with eigenvalues written with rest masses mi ) and of an external Hamiltonian Hext : Hext =

1 −−→ ⇒ m −−→ ⇒ − → −−→ −−→ → − −−→ pop . g (t).− p−→ qop . γ (t).− q−→ op − op − Ω(t).(qop × pop ) − m g (t).qop , 2m 2

(1)

− →

→ where one recognizes several usual gravito-inertial effects: rotation in Ω(t), gravity in − g (t), gradient of gravity in ⇒ ⇒ γ (t), . . . and where g (t) is usually taken equal to the unity tensor in the absence of gravitational wave. For a wave packet ψ(q, t1 ) = wp(t1 , q −q1 , p1 , X1 , Y1 ), where q1 is the initial mean position of the wave packet, p1 its initial mean momentum, and (X1 , Y1 ) its initial complex width parameters in phase space, one obtains the ABCDξ theorem [2]:  ψ(q, t2 ) = d 3 q  .K(q, t2 , q  , t1 ).wp(t1 , q  − q1 , p1 , X1 , Y1 ) i

= e h¯ Scl (t2 ,t1 ,q1 ,p1 ) .wp(t2 , q − q2 , p2 , X2 , Y2 ),

(2)

where K and Scl are, respectively, the quantum propagator and the classical action, and where q2 , p2 , X2 , Y2 obey ⇒ the ABCD law (G and R are the representative matrices of g (t) and of the rotation operator corresponding to − → Ω(t), and we write A21 instead of A(t2 , t1 ) for simplicity):         q2 A21 B21 q1 R21 .ξ21 (3) + · = , ˙ p2 /m G−1 C21 D21 p1 /m 2 .R21 .ξ21       A21 B21 X1 X2 = . . (4) Y2 C21 D21 Y1 For example, the phase of a Gaussian wave packet is (for simplicity we shall omit the transposition sign ∼ on matrix representations of vectors): Scl (t2 , t1 , q1 , p1 )/h¯ + p2 .(q − q2 )/h¯ +

  m (q − q2 ). Re Y2 .X2−1 .(q − q2 ) 2h¯

(5)

and in this case the main phase shift between t1 and t2 is equal to:

Scl (t2 , t1 , q1 , p1 )/h¯ + p1 .q1 /h¯ − p2 .q2 /h¯ .

(6)

3. The ttt theorem When the dispersive nature of a beam splitter is neglected (i.e., the wave packet structure is preserved), its effect may be summarized by the introduction of both a phase and an amplitude factor (see [13] and [4] for a detailed proof): Mba .e−i(ω

∗ t ∗ −k ∗ .q ∗ +ϕ ∗ )

,

(7)

where t ∗ and q ∗ depend on tA and qA , the mean time and position of the electromagnetic wave used as a beam splitter.

Ch. Antoine, Ch.J. Bordé / Physics Letters A 306 (2003) 277–284

279

For a temporal beam splitter: t ∗ ≡ tA , q ∗ ≡ qcl (tA ), k ∗ ≡ k, ω∗ = ω, ϕ ∗ ≡ ϕ (laser phase).

(8)

For a spatial beam splitter: q ∗ ≡ qA ,   t ∗ such that qcl t ∗ ≡ qA , k ∗ ≡ k + δk, ω∗ = ω, ϕ ∗ ≡ ϕ + δϕ,

(9)

where qcl is the central position of the input atomic wave packet (equal to the classical position because of Ehrenfest theorem), where δk is the additional momentum transferred to the excited atoms out of resonance, and where δϕ is a laser phase: δϕ ≡ −δk.qA (see [4]). Let us emphasize that these calculations do not rely on the assumption that the splitter is infinitely thin or that the atom trajectories are classical.

4. The four end-points theorem for a Hamiltonian at most quadratic in position and momentum operators We shall cut any interferometer into as many slices as there are interactions on either arm and thus obtain several path pieces (see Section 5). From now on we shall consider systematically pairs of these homologous paths (see Fig. 1) in the case of a Hamiltonian at most quadratic. These two classical trajectories are labelled by their corresponding mass (mα and mβ ), their initial position and momentum (qα1 , pα1 , qβ1 and pβ1 ) and their common drift time T = t2 − t1 . Before establishing the first new theorem let us consider the expression of the classical action for the α path (see [2]): Scl (t2 , t1 , qα1 , pα1 ) mα  = ξ˙ .R.G

−1

t2 .(A.qα1 + B.pα1 /mα ) + t1

→ −

  pα1 pα1 L AC pα1 BD  α1 , .qα1 + . dt + qα1. . + .BC.q mα 2 mα 2 mα mα

(10)

− →

where ξ and L depend on g (t) (see [2] for notations). This expression can be rewritten as: Scl (t2 , t1 , qα1 , pα1 ) mα pα2 pα1 1  −1 = .qα2 − .qα1 − ξ˙ .R.G .R.ξ + 2mα 2mα 2

t2 t1

1  −1 1  pα2 L dt + ξ˙ .R.G .qα2 − ξ.R. mα 2 2 mα

(11)

280

Ch. Antoine, Ch.J. Bordé / Physics Letters A 306 (2003) 277–284

Fig. 1. A pair of homologous paths.

with the help of the definition of qα2 and with:

pα2 mα

 −1 (see formula (3)). Then we can use the β path to replace ξ˙ .R.G

 −1 = pβ2 − C.qβ1 − D. pβ1 . ξ˙ .R.G mβ mβ

(12)

Consequently we get:     Scl (t2 , t1 , qα1 , pα1 ) 1 pα2 pβ2 1 pα1 pβ1 = + + .qα2 − .qα1 + h(t2 , t1 ) + f (α, β), mα 2 mα mβ 2 mα mβ

(13)

where h(t2 , t1 ) is independent of positions and momenta and where f (α, β) = f (β, α). The same conclusion holds for the expression of Scl (t2 , t1 , qβ1 , pβ1 )/mβ which is obtained by exchanging α and β. Finally, we arrive at the first new theorem (a more general demonstration starting with Hamilton principal functions is given in Appendix A): Theorem 1.     1 pα1 pβ1 Scl (t2 , t1 , qα1 , pα1 ) 1 pα2 pβ2 .qα2 + .qα1 − + + mα 2 mα mβ 2 mα mβ     Scl (t2 , t1 , qβ1 , pβ1 ) 1 pα2 pβ2 1 pα1 pβ1 .qβ2 + .qβ1 = − + + mβ 2 mα mβ 2 mα mβ

(14)

or equivalently:   pβ1 Scl (t2 , t1 , qβ1 , pβ1 ) pβ2 Scl (t2 , t1 , qα1 , pα1 ) pα2 pα1 − .qα2 + .qα1 − − .qβ2 + .qβ1 mα mα mα mβ mβ mβ       pβ2 pα2 pβ1 pα1 qα2 + qβ2 qα1 + qβ1 = − . . − − mβ mα 2 mβ mα 2

(15)

which will give the main part of the phase shift expressed with the half sums of the coordinates and the momenta of the four end-points only. In the case of identical masses (mα = mβ ) this expression simplifies to:

Scl (t2 , t1 , qα1, pα1 ) − pα2 .qα2 + pα1 .qα1 − Scl (t2 , t1 , qβ1 , pβ1 ) − pβ2 .qβ2 + pβ1 .qβ1     qα2 + qβ2 qα1 + qβ1 = (pβ2 − pα2 ). − (pβ1 − pα1 ). . 2 2

(16)

Ch. Antoine, Ch.J. Bordé / Physics Letters A 306 (2003) 277–284

281

5. The phase shift formula for a Hamiltonian at most quadratic in position and momentum operators In this section we draw on the results of previous sections to establish the interferometer phase shift expression for an arbitrary beam splitters configuration. For a sequence of pairs of homologous paths (an interferometer geometry) (see Fig. 2) one can infer the general sum for the main coordinate-dependent part of the global phase shift:   N qαD + qβD pα1 + pβ1 pβD − pαD qαi + qβi . q− − .(qβ1 − qα1 ) + . (17) (kβi − kαi ). h¯ 2 2h¯ 2 i=1

If now we take into account the other terms of the phase shift, we finally obtain the following result (given here for a Gaussian wave packet): Theorem 2. +φ(q, tN+1 ≡ tD )

  pα1 + pβ1 qαD + qβD /h¯ − .(qβ1 − qα1) = (pβD − pαD ). q − 2 2h¯  N  qαi + qβi − (ωβi − ωαi ).ti + ϕβi − ϕαi + (kβi − kαi ). 2 i=1

   N  mβi − mαi Sαi pα,i+1 pαi + h¯ kαi . + .(qβ,i+1 − qα,i+1 ) − .(qβi − qαi ) 2h¯ mαi 2mαi 2mαi i=1   pβ,i+1 pβi + h¯ kβi Sβi + .(qα,i+1 − qβ,i+1 ) − .(qαi − qβi ) + mβi 2mβi 2mβi    mβ,N m α,N −1 −1  (q − qβD ). Re YD .XD (q − qαD ). Re YD .XD + .(q − qβD ) − .(q − qαD ), 2h¯ 2h¯ ≡ Scl (ti+1 , ti , qαi , pαi + h¯ kαi , mαi ). +

where Sαi

(18)

Fig. 2. Interferometer geometry sliced into pairs of homologous paths between interactions on either arm (when an interaction occurs only on one arm the corresponding k on the other arm is set = 0).

282

Ch. Antoine, Ch.J. Bordé / Physics Letters A 306 (2003) 277–284

This basic formula is valid for a time-dependent Hamiltonian and takes into account all the mass differences which may occur. It allows to calculate exactly the phase shift for all the interferometer geometries which can be sliced as above: symmetrical Ramsey–Bordé (Mach–Zehnder), atomic fountain clocks,. . . . All these particular cases will be detailed in forthcoming papers (see [6]). Let us point out that the nature (temporal or spatial) of beam splitters leads to different slicing of the paths. In the spatial case, indeed, the number of different ti∗ may be twice as great as in the temporal case (see the definition of ti∗ in these two different cases in Section 3).

6. Phase shift after spatial integration In any interferometer one has to integrate spatially the output wave packet over the detection region. With Gaussian wave packets this integration leads to a mid-point theorem [3,6]: “The first term of +φ(q, tD ) disappears when the spatial integration is performed”. Furthermore the terms which depend on the wave packets structure (Y and X) vanish when mβ,N = mα,N (which is always the case). One obtains finally: pα1 + pβ1 .(qβ1 − qα1 ) 2h¯  N  qαi + qβi + − (ωβi − ωαi ).ti + ϕβi − ϕαi (kβi − kαi ). 2

+φ(tD ) = −

i=1

   N  mβi − mαi Sαi pα,i+1 pαi + h¯ kαi . + .(qβ,i+1 − qα,i+1 ) − .(qβi − qαi ) + 2h¯ mαi 2mαi 2mαi i=1   pβ,i+1 pβi + h¯ kβi Sβi + + .(qα,i+1 − qβ,i+1) − .(qαi − qβi ) . mβi 2mβi 2mβi (19)

7. Identical masses and symmetrical case The case of identical masses is an important approximation which is commonly used for the modelization of many devices like gravimeters and gyrometers [7–9]. If mαi = mβi = m, ∀i, this general phase shift becomes: +φ(tD ) = −



pα1 + pβ1 qαi + qβi .(qβ1 − qα1 ) + + (kβi − kαi ). ϕβi − ϕαi − (ωβi − ωαi ).ti . 2h¯ 2 N

N

i=1

i=1

(20)

We can also specify the form of this phase shift when the interferometer geometry is symmetrical (see Fig. 3). This symmetry is expressed as: kβi + kαi = 0, ∀i ∈ [2, N − 1], i.e., it is a symmetry with respect to the direction of the particular vector: pinitial + h¯ kinitial/2. Consequently: +φ(tN ) = k1 .q1 + 2

N−1 i=2

qαi + qβi qαN + qβN + kN . − (ϕβi − ϕαi ). 2 2 N

ki .

i=1

(21)

Ch. Antoine, Ch.J. Bordé / Physics Letters A 306 (2003) 277–284

283

Fig. 3. A typical symmetrical interferometer.

But ∀i ∈ [1, N − 1]: qαi + qβi qα,i+1 + qβ,i+1 Bi+1,i pαi + pβi = ξi+1,i + Ai+1,i . + . 2 2 2 m  Bi+1,1 h¯ k1 ≡ Q(ti+1 ) . p1 + = ξi+1,1 + Ai+1,1 .q1 + m 2 which can be calculated with the ABCDξ law. ¯ 1 (“Bragg initial momentum”). It depends only on q1 (“initial position”) and p1 + hk 2 Therefore: +φ(tN ) =

N N (kβi − kαi ).Q(ti ) − (ϕβi − ϕαi ) i=1

(22)

(23)

i=1

which has a very simple form when the Bragg condition p1 +

h¯ k1 2

= 0 is satisfied.

8. Conclusion In this Letter we have used the ABCDξ formulation of atom optics and the ttt theorem to establish two theorems valid for a time-dependent Hamiltonian at most quadratic in position and momentum operators. The first one gives a compact expression of the action difference between two homologous paths. The second one gives an analytical expression of the global phase shift for atom interferometers in the case of such a Hamiltonian. Consequently this analytical expression provides a simple way to calculate exactly the phase shift in this case, and then one can calculate perturbatively the effect of a higher-order term in the external Hamiltonian (necessary for space missions like HYPER [10]). For example, one can calculate exactly the global phase shift due to gravity plus a gradient of gravity plus a rotation, and then calculate perturbatively the effect of a gradient of gradient of gravity. These calculations and the application to specific cases (gravimeters, gyrometers, atomic clocks, . . . ) will be detailed in a forthcoming article [6] where we recover well-known perturbative results ([5,9,11,12]) from exact expressions. Appendix A In the case of a Hamiltonian at most quadratic in position and momentum operators, the Hamilton principal functions concerning two pairs of homologous points are also at most quadratic in positions (owing to the

284

Ch. Antoine, Ch.J. Bordé / Physics Letters A 306 (2003) 277–284

Hamilton–Jacobi equation, see [2]): Sα (qα1 , qα2)/mα = a + b.qα1 + c.qα2 + qα1 .d.qα1 + qα1 .e.qα2 + qα2 .f.qα2,

(A.1)

Sβ (qβ1 , qβ2 )/mβ = a + b.qβ1 + c.qβ2 + qβ1 .d.qβ1 + qβ1 .e.qβ2 + qβ2 .f.qβ2 ,

(A.2)

where a is a scalar, b and c are vectors, and d, e and f are matrices (see [2]). We can define pα1 , pα2 , pβ1 , pβ2 such that   pα1 Sα = −b − 2d.qα1 − e.qα2, ≡ −∇qα1 mα mα   pα2 Sα ≡ ∇qα2 ˜ α1 , = c + 2f.qα2 + e.q mα mα   pβ1 Sβ = −b − 2d.qβ1 − e.qβ2, ≡ −∇qβ1 mβ mβ   Sβ pβ2 ≡ ∇qβ2 ˜ β1 = c + 2f.qβ2 + e.q mβ mβ and obtain the following expression:     Sβ Sα 1 pα2 pβ2 1 pα1 pβ1 − = + + .(qα2 − qβ2 ) − .(qα1 − qβ1 ). mα mβ 2 mα mβ 2 mα mβ

(A.3) (A.4) (A.5) (A.6)

(A.7)

The same relation holds for the classical action concerning two actual paths with a common drift time (homologous paths). This yields an other demonstration of the first theorem expressed in Section 4.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

P. Berman (Ed.), Atom Interferometry, Academic Press, San Diego, 1997. Ch.J. Bordé, C. R. Acad. Sci. Paris Série IV 2 (2001) 509. Ch.J. Bordé, Metrologia 39 (5) (2002) 435. Ch.J. Bordé, An elementary quantum theory of atom-wave beam splitters: the ttt theorem, Lecture Notes for a Mini-Course, Institut für Quantenoptik, Universität Hannover, 2002, to be published. J. Audretsch, K.-P. Marzlin, J. Phys. II (France) 4 (1994) 2073. Ch. Antoine, Ch.J. Bordé, J. Opt. B, submitted for publication. A. Peters, K.Y. Chung, S. Chu, Metrologia 38 (2001) 25. M.J. Snadden, J.M. McGuirk, P. Bouyer, K.G. Haritos, M.A. Kasevich, Phys. Rev. Lett. 81 (1998) 971. P. Wolf, Ph. Tourrenc, Phys. Lett. A 251 (1999) 241. R. Bingham, et al., HYPER, Hyper-precision cold atom interferometry in space, Assessment Study report, ESA-SCI (2000). Ch.J. Bordé, Phys. Lett. A 140 (1989) 59. Ch.J. Bordé, Atomic interferometry and laser spectroscopy, in: Laser Spectroscopy X, World Scientific, Singapore, 1991, pp. 239–245. J. Ishikawa, F. Riehle, J. Helmcke, Ch.J. Bordé, Phys. Rev. A 49 (1994) 4794.